and we use the notation, \(\eta _t(f)=\int _{\mathbb {R}^{r_1}}f(x)\eta _t(dx)\)  for \(f:\mathbb {R}^{r_1}\rightarrow \mathbb {R}^{r}\)  that is \(\eta _t-\) integrable (in particular, \(r=r_1\)  in () and () and \(r=r_1^2\)  in (REF )).
Any of these probabilistic models will be commonly referred to as Kalman-Bucy (nonlinear) diffusion processes. In the following we will denote by \({\mathcal {G}}_t\)  the augmented filtration generated by \(\overline{X}_0\)  and the triplet of independent Brownian motions \((Y_t,\overline{W}_t,\overline{V}_t)\) . The process (REF ) corresponds to the vanilla type ensemble
Kalman-Bucy filter that is typically used in the literature. The process () is associated the so-called deterministic ensemble Kalman-Bucy filter [1]} and () is a deterministic transport-inspired equation [2]}.
